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Lithium diisopropylamide is a strong, sterically hindered base used to generate enolates. Its low nucleophilicity toward many carbonyl substrates helps separate deprotonation from carbonyl addition.

Number the cyclopentenone carbonyl carbon $C_1$, its alkene carbons $C_2$ and $C_3$ with OR at $C_3$, and its saturated carbons $C_4$ and $C_5$. The ester alkene has alpha carbon $X$, bearing Me and CO2Et, and beta carbon $Y$, bearing the protected carbon substituent. lithium diisopropylamide generates the ordinary ketone enolate at the saturated alpha carbon $C_5$. The acetonide is retained.

The carbon end of this enolate attacks ester beta carbon Y. Its pi system shifts through X onto the ester oxygen, forming the first new bond $C_5-Y$ and an ester enolate at X. That enolate now attacks the original cyclopentenone beta carbon $C_3$ intramolecularly, giving the second new bond $X-C_3$. The original enone electrons move onto its ketone oxygen, and protonation of the final ketone enolate completes the product. The OR substituent is retained at the new bridgehead; it need not leave in either Michael addition.

The atom map provides a useful structural check. Between bridgeheads $C_3$ and $C_5$, the three paths are $C_3-C_2-C_1-C_5$, $C_3-C_4-C_5$, and $C_3-X-Y-C_5$. Their numbers of internal atoms are 2, 1 and 2, establishing the norbornane skeleton. Both original carbonyls survive and both initial alkenes are consumed. The two successive conjugate additions give this atom map without requiring a concerted cycloaddition.

The first base equivalent removes O–H; the second removes the alpha C–H. The first equivalent of lithium diisopropylamide converts the carboxylic acid to its lithium carboxylate. A further equivalent removes the remaining hydrogen on the carbon between the two allyl groups, giving a carboxylic acid dianion:The second negative charge has enolate-like delocalization into the carboxyl group; lithium coordination and aggregation are not represented by this simple charge-bookkeeping structure. The carbon nucleophile attacks methyl iodide with C–I bond cleavage, forming the new C–Me bond. acid workup reprotonates the carboxylate, giving $F=\mathrm{H}{\mathrm{O}}_2\mathrm{C}-\mathrm{C}(\mathrm{Me})(\mathrm{L}{)}_2$. The two equivalents are therefore chemically necessary for the stated carbon alkylation rather than simply an arbitrary excess. A single equivalent is consumed predominantly by the much more acidic carboxylic proton. The product has formula ${\mathrm{C}}_9{\mathrm{H}}_{14}{\mathrm{O}}_2$ and still has two identical allyl groups, so this methylation does not create a stereocenter.

Lithium diisopropylamide removes an alpha proton from the ethyl side of the ketone. The C–H electrons form the C=C part of a lithium enolate, while the C=O pi electrons move to oxygen. The phenyl side has no competing alpha C–H site. Diisopropylamine is the conjugate-acid byproduct.The carbon end of this enolate attacks the primary carbon bearing iodine in $\mathrm{IC}{\mathrm{H}}_2\mathrm{C}{\mathrm{H}}_2\mathrm{C}{\mathrm{H}}_2\mathrm{OMe}$. The C–I bond breaks to iodide in an SN2 reaction, and carbonyl reformation givesThus a three-carbon methoxypropyl chain is attached at the ketone alpha carbon. The remote methyl ether survives. The hindered amide base is chosen to generate the enolate rather than to add to the carbonyl. No absolute stereochemical preference is specified for the new stereocenter under these achiral conditions.

The target is 5-Aminohexanoic acid, $\mathrm{H}{\mathrm{O}}_2\mathrm{C}(\mathrm{C}{\mathrm{H}}_2{)}_3\mathrm{CH}(\mathrm{N}{\mathrm{H}}_2)\mathrm{Me}$. Disconnect its terminal amine/carboxylic-acid relationship through hydrolysis of 6-methylpiperidin-2-one. This six-membered lactam follows from a regioselected Beckmann ring expansion of 2-methylcyclopentanone oxime.

Start with cyclopentanone. Form its enolate with lithium diisopropylamide in dry THF at low temperature, add methyl iodide, then quench to obtain 2-methylcyclopentanone. Control the methyl iodide/base stoichiometry to suppress a second alkylation. React the ketone with hydroxylamine to form its oxime and select the geometrical oxime isomer in which the bond to the methyl-substituted adjacent ring carbon is anti to the departing oxime oxygen group.

Activate the oxime, for example under appropriate acid-promoted Beckmann conditions. The anti methyl-bearing C–C bond migrates onto nitrogen as the N–O-derived leaving group departs, inserting nitrogen between the original carbonyl carbon and the methyl-substituted ring carbon. Hydration/tautomerization gives 6-methylpiperidin-2-one. The opposite oxime geometry would insert nitrogen on the other side and ultimately place methyl at a different position in the amino acid; oxime geometry is therefore an essential part of this proposal.

Hydrolyse the lactam with aqueous acid and heat, then adjust the work-up to the desired acid/ammonium form. Cleavage of its acyl–nitrogen bond opens the ring to $\mathrm{H}{\mathrm{O}}_2\mathrm{C}(\mathrm{C}{\mathrm{H}}_2{)}_3\mathrm{CH}(\mathrm{N}{\mathrm{H}}_2)\mathrm{Me}$ after suitable neutralization. All five original cyclopentanone carbons are retained, and the one methyl carbon comes from methyl iodide. A racemic mixture product is appropriate to the stereochemically unspecified target. The reaction sequence is an organic ring-expansion synthesis and requires no biological transformation.